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Home > PC > Chapter 4 > Lesson 4.1.4 > Problem 4-53


For each graph below, find an equation using sine and an equation using cosine that will generate the graph. For each graph, one of the functions you find should not require a horizontal shift.

  1. Periodic curve, x axis scaled from 0 to 3 pi, with 3 visible turning points at (1 half pi, comma 1), (3 halves pi, comma negative 5), & (5 halves pi, comma 1), passing through the points (pi, comma negative 2), & (2 pi, comma negative 2).

  1. Periodic curve, x axis scaled from 0 to 3 pi, with 4 visible turning points at (0, comma negative 1), (pi, comma 3), (2 pi, comma negative 1), & (3 pi, comma 3), passing through the points (1 half pi, comma 1), (3 halves pi, comma 1) & (5 halves pi, comma 1).

Equation: y = Capital A, sine of capital F, open parenthesis, small x, plus or minus capital H, close parenthesis, plus or minus capital V, arrows pointing and labeled as follows, to capital A, amplitude, distance from middle to top, arrow to capital F, Frequency, this is not period, P, P F = 2 pi, arrow to Capital H, horizontal shift, the sign is opposite the directions. arrow to capital V, vertical shift, the whole curve is moved up or down in the direction of the sign.

Added to periodic curve, 3 horizontal lines at, y = negative 5, y = negative 2, & y = 1.

Added: vertical up arrow at x = 3 halves pi, from middle horizontal to top horizontal.

Added: horizontal double sided arrow from (0, comma negative 2), to (2 pi, comma negative 2).

Added: vertical down arrow at x = 1 half pi, from top horizontal to middle horizontal.

Draw a horizontal line through the center, top, and bottom of the curve.

The amplitude is from the middle up to the top of the curve.

The period is the length of one cycle. In this case it is , so .

Shifts: Horizontal (left or right shift from -axis). Vertical (up or down shifts from -axis.)

Use the eTool below to visualize this problem.
Click on the link to the right to view the full version of the eTool: PCT 4-53 eTool